Matter correlators through a wormhole in double-scaled SYK

We compute the two-point function of matter operators in the double-scaled SYK (DSSYK) model, where the two matter operators are inserted at each end of the cylindrical wormhole. We find that the wormhole amplitude in DSSYK is written as a trace over the chord Hilbert space. We also show that the length of the wormhole is stabilized in the semi-classical limit, by the same mechanism worked for the JT gravity case.


Introduction
Wormhole geometries have played an important role in the recent progress of our understanding of quantum black holes and the Page curve [1,2]. 1 However, in the AdS/CFT correspondence, the existence of a wormhole connecting disjoint boundaries of bulk spacetime leads to a puzzle, known as the factorization problem [17].At least in lower dimensional examples of quantum gravity, such as the JT gravity and the SYK model, this factorization problem is naturally solved by assuming that the boundary theory involves an ensemble average over the random Hamiltonians.Indeed, as shown in [18] the JT gravity is holographycally dual to a random matrix model, where the random matrix plays the role of random Hamiltonians.
According to the classification in [4], there are three types of averaging in the boundary theory: (i) averaging over Hamiltonians, (ii) averaging over operators, and (iii) averaging over states.In this paper, we consider the wormhole amplitude in the double-scaled SYK (DSSYK) model [19,20].In this case, the wormhole originates from the hybrid of averagings (i) and (ii) above.We compute the two-point function of matter operators in DSSYK, where the two matter operators are inserted at each end of the cylindrical wormhole.Our computation is a natural generalization of the similar computation for JT gravity in [3].We find that the wormhole amplitude in DSSYK is written as a trace over the chord Hilbert space, which was interpreted in [21] as a Hilbert space of the bulk gravity theory.Moreover, we find that the length of the wormhole is stabilized in the semi-classical limit, by the same mechanism worked for the JT gravity case [3].
This paper is organized as follows.In section 2, we review the result of DSSYK and its two-matrix model representation, known as the ETH matrix model.In section 3, we compute the two-point function of matter operators on the wormhole geometry.In section 4, we study the saddle-point approximation of the wormhole amplitude and confirm that the length of the wormhole is stabilized at a non-zero length.Finally we conclude in section 5 with some discussion on the future problems.In appendix A we consider the JT gravity limit of the wormhole amplitude.

Review of DSSYK
In this section we briefly review the result of DSSYK in [20] and the ETH matrix model for DSSYK in [22].
The SYK model is defined by the Hamiltonian for M Majorana fermions where J i 1 •••ip is a random coupling drawn from the Gaussian distribution.DSSYK is defined by the scaling limit As shown in [20], the ensemble average of the moment Tr H k reduces to a counting problem of the intersection number of chord diagrams with q = e −λ .This counting problem is solved by introducing the transfer matrix T where A ± denotes the q-deformed oscillator acting on the chord number state |n⟩ Then the disk amplitude of DSSYK is given by ⟨Tr e −βH ⟩ J = ⟨0|e βT |0⟩. (2.6) The transfer matrix T becomes diagonal in the θ-basis2 and the overlap of ⟨n| and |θ⟩ is given by the q-Hermite polynomial H n (cos θ|q) ⟨n|θ⟩ = H n (cos θ|q) where the q-Pochhammer symbol is defined by (a i ; q) n . (2.9) |θ⟩ and |n⟩ are normalized as and the measure factor µ(θ) is given by µ(θ) = (q, e ±2iθ ; q) ∞ . (2.11) Then the disk partition function (2.6) is written as θ) . (2.12) As discussed in [20], we can also consider the matter operator with Gaussian random coefficients K i 1 •••is which is drawn independently from the random coupling J i 1 •••ip in the SYK Hamiltonian.In the double scaling limit (2.2), the effect of this operator can be made finite by taking the limit s → ∞ with ∆ = s/p held fixed.Then the correlator of O's is also written as a counting problem of the chord diagrams Note that there appear two types of chords in this computation: H-chords and O-chords coming from the Wick contraction of random couplings The O-chord is also called the matter chord.For instance, the disk two-point function of O is given by ⟨Tr where N is the number operator In terms of the θ-basis, the disk two-point function in (2.15) is written as In the rest of this paper, we will ignore the intersections between the matter chords for simplicity.This amounts to consider a bulk free field which is holographically dual to the matter operator O.
Next, let us consider the ETH matrix model for DSSYK introduced in [22].As discussed in [22], we can define the two-matrix model which reproduces the disk partition function (2.6) and the two-point function (2.15) of DSSYK in the large N limit.The N × N matrices A and B correspond to H and O, respectively, and the size N of the matrices is given by the dimension of the Fock space of M Majorana fermions (2.20) The potential V (A, B) in (2.19) is given by [22] Tr where T n (cos θ) = cos(nθ) denotes the Chebyshev polynomial of the first kind and E a is the eigenvalue of the matrix A given by E(θ a ) in (2.7).The first line of (2.21) is determined by requiring that the eigenvalue density of the matrix A reproduces the measure factor µ(θ) in (2.11) in the large N limit.The second line of (2.21) is constructed in such a way that the two-point function of the matrix B reproduces the disk two-point function of O in (2.17).From (2.21), we can read off the propagator of the matrix B as where the expectation value is taken in the two-matrix model We can check that the disk two-point function of O in (2.17) is reproduced from the twomatrix model (2.24) Here we have used the relation which is valid in the large N limit.In the next section, we compute the matter correlator of DSSYK on the wormhole geometry using the propagator in (2.22).

Matter correlators through a wormhole
Let us consider the matter correlator of DSSYK on the wormhole geometry.In the twomatrix model description of DSSYK, it is given by Tr(e −β L A B) Tr(e −β R A B) .Using the propagator of B in (2.22) we find Note that in this computation only the a = b terms survive, which is reminiscent of the Berry's "diagonal approximation" for the spectral form factor [23].Note also that the power of N in (3.1) is down by one compared to the disk two-point function in (2.24).Using (2.10) and (2.18), we can easily show that (3.1) is written as where Tr H denotes the trace over the Hilbert space H spanned by the chord number states More explicitly, the trace in (3.2) is written as This amplitude represents the cylindrical wormhole which is schematically depicted as where the circumference β of the cylinder is given by In other words, the amplitude in (3.2) represents the correlation of matter operators through the wormhole.Our computation is a natural generalization of the similar computation in the JT gravity [3].In particular, the operator q ∆ N in (3.2) is a discrete analogue of the operator G O appeared in [3] (see also [24,25]) where ℓ denotes the length of the bulk geodesic connecting the two operators O.Note that ℓ can be negative since ℓ is a renormalized geodesic length.We can insert more operators O on the two boundaries of the cylinder For instance, when k = 2 (3.9) becomes (3.10)

Stabilization of the wormhole length
In general, the wormhole geometry with a fixed length is not a classical solution of the bulk gravity theory.However, as discussed in [3] for the JT gravity case, the length of the wormhole is stabilized by including the effect of matter correlator.This is similar in spirit to the construction of wormhole solutions as gravitational constrained instantons [6,7].Below we will see that the length n of the wormhole in (3.4) is stabilized in the semi-classical saddle-point approximation.
The wormhole amplitude in (3.4) is written as where β is defined in (3.6).Let us consider the saddle-point approximation of the θ-integral in the semi-classical limit λ → 0. Using the relation the measure factor µ(θ) is approximated as Here Li 2 (z) denotes the dilogarithm function.Also, using the relation the factor |⟨n|θ⟩| 2 in (4.1) is written as the integral in (4.1) is written as where We can easily show that the solution of the saddle-point equation is given by ϕ * = 0, sin 2 θ * = q n .(4.10) Then E(θ) in (4.1) is approximated as Alternatively, the same approximation can be obtained from the recursion relation obeyed by From this relation, cos θ is approximated as which agrees with the result of saddle-point approximation (4.10).We should stress that our saddle-point analysis is different from [26].In [26], the βE(θ) term is assumed to be of order O(λ −1 ) and hence the saddle-point value of θ depends on β.In our case we assumed the scaling in (4.6), which implies that βE(θ) is sub-leading compared to the leading term λ −1 F (θ, ϕ) in (4.7).Thus the βE(θ) term does not enter into the saddle-point equation (4.9).Finally, we arrive at the saddle-point approximation of the wormhole amplitude where (4.15) In Figure 1, we show the plot of S eff (n) in (4.15).We can see that S eff (n) has a minimum at a non-zero value of λn and hence the wormhole length n is stabilized by the effect of matter correlator.This stabilization mechanism of the wormhole length for DSSYK is essentially the same as that for the JT gravity discussed in [3] (see appendix A for the JT gravity limit of the wormhole amplitude (4.1)).From (4.15), we find that the solution n = n * of the saddle-point equation ∂ n S eff (n) = 0 is given by where β is defined by β We can also estimate the variance of the wormhole length from the second derivative of As discussed in [3], the β → 0 limit of the wormhole amplitude in JT gravity is UV divergent since the cylinder becomes very thin as β → 0. Interestingly, in the case of DSSYK the wormhole amplitude in (3.2) is finite in this limit lim β→0 Tr H e βT q ∆ N = Tr Thus, DSSYK can be thought of as a UV completion of the JT gravity.As discussed in [21,22,27,28], the UV finiteness of DSSYK is closely related to the existence of a minimal length in the bulk geometry.Note that the wormhole solution with a finite length ceases to exist in the thin cylinder limit β → 0 since the saddle-point solution in (4.16) collapses to a zero-length wormhole n * = 0.In other words, the semiclassical approximation breaks down in this limit.Instead, we have to sum over all n as shown in (4.19).

Conclusion and outlook
In this paper we have studied the matter two-point function through the wormhole geometry in DSSYK.We find that this two-point function (3.2) is written as a trace on the Hilbert space spanned by the chord number states, which can be identified as the Hilbert space of the bulk gravity theory [21].In the saddle-point approximation, the sum over n in (3.4) is dominated by a non-zero value of n due to the balance between the matter contribution q ∆n and the Boltzmann factor e −βE(θ * ) in (4.14).This stabilization mechanism of the wormhole length is the same as that appeared in the computation of the wormhole amplitude in JT gravity [3].This is similar in spirit to the gravitational constrained instantons in [6,7].One important difference from [3] is that the wormhole amplitude (3.2) in DSSYK is finite even in the thin cylinder limit (4.19).There are many interesting open questions.In this paper, we have ignored the intersections of matter chords, which amounts to consider the free field in the bulk theory.It would be interesting to incorporate the matter interaction and see how the result in this paper is modified.It would also be interesting to generalize our computation to higher genus topologies along the lines of [29,30].The higher dimensional generalization of our computation would be also interesting (see e.g.[8] for a wormhole in the AdS 3 /CFT 2 correspondence).
We have seen that the stabilization mechanism of the wormhole length advocated in [3] works for DSSYK as well.However, we should stress that the existence of a semi-classical saddle in the gravitational path integral is not a necessary condition for the definition of quantum gravity.In our case, the wormhole length n is a dynamical variable and the wormhole amplitude (3.4) is defined by summing over all possible n.The wormhole amplitude (3.4) itself is well-defined without requiring the existence of a semi-classical saddle.In fact, the wormhole amplitude of DSSYK is finite in the thin cylinder limit (4.19), which is far from the semi-classical regime.By definition, quantum gravity should make sense in the quantum regime, which is indeed the case in our computation of the wormhole amplitude in DSSYK.

Figure 2 :
Figure 2: Plot of s sinh(2πs)|K 2is (2x)| 2 as a function of s for a fixed x.The horizontal axis is s/x.In this figure we set x = 20.